I understand the following:

Theorem: Suppose the real functions and are continuous and have continuous 1st order partials in a domain D. If and satisfy the CR equations at all points in D, then is analytic in D.

Therefore, the CR equations and must hold.

But I don't understand what I'm supposed to conclude when I do the following problem:

When I do the CR equations I get and . Does that mean it is only differentiable at ? What does that say about being analytic? I've seen a few problems like this were some of the CR equations are equal to zero such as . For that one I understand that limits at different paths are not equal therefore only differentiable at . Do I need to do different paths to show where it's only differentiable for the problem above or how should I try to tackle this problem? Thank you so much!